Breast cancer screening policies attempt to achieve timely diagnosis by regularly screening healthy women via various imaging tests. Various clinical decisions are needed to manage the screening process: selecting initial screening tests, interpreting test results, and deciding if further diagnostic tests are required. Current screening policies are guided by clinical practice guidelines (CPGs), which represent a 'one-size-fits-all' approach, designed to work well (on average) for a population, and can only offer coarse expert-based patient stratification that is not rigorously validated through data. Since the risks and benefits of screening tests are functions of each patient's features, personalized screening policies tailored to the features of individuals are desirable. To address this issue, we developed ConfidentCare: a computer-aided clinical decision support system that learns a personalized screening policy from electronic health record (EHR) data. By a 'personalized screening policy,' we mean a clustering of women's features, and a set of customized screening guidelines for each cluster. ConfidentCare operates by computing clusters of patients with similar features, then learning the 'best' screening procedure for each cluster using a supervised learning algorithm. The algorithm ensures that the learned screening policy satisfies a predefined accuracy requirement with a high level of confidence for every cluster. By applying ConfidentCare to real-world data, we show that it outperforms the current CPGs in terms of cost efficiency and false positive rates: a reduction of 31% in the false positive rate can be achieved.